A black hole in Einstein-Maxwell theory can have the NUT parameter in addition to the well known hairs of black hole, namely the mass \(m\) , rotation \(a\) , and electric charge \(q\)  [1]}. Sometime adding this NUT parameter to a known black hole solution in theory can be considered as a sort of NUT extension to the solution. For example, adding NUT parameter to the Schwarzschild spacetime yields the Taub-NUT metric of the vacuum Einstein system. One can also add NUT parameter to the Kerr-Newman solution to yield the Kerr-Newman-Taub-NUT spacetime. The Taub-NUT solution mentioned previously is just a neutral and static limits of Kerr-Newman-Taub-NUT family. Despite the obscure realization of NUT parameter in our real world, solutions containing this quantity have contributed significantly in the gravitational studies. For example the motion of charged test particle was studied in [2]}, and the gravitomagnetism in this spacetime was investigated in [3]}. Spacetime with NUT parameter can also exist in gravitational theories beyond Einstein-Maxwell, for example in scalar-tensor [4]}, Randall-Sundrum braneworld [5]}, and low energy heterotic string [6]} theories. In a recent work [7]}, the authors show that the Misner string contribution to the Taub-NUT-AdS entropy can be renormalized by introducing the Gauss-Bonnet term.
